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What if we have two predictor variables? We want to predict depression. We have measured stress and loneliness. We can ask several questions: 1) which is the stronger predictor? 2) how well do they predict depression together? 3) what is the effect of loneliness on depression, controlling for stress?

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What if we have two predictor variables? Regressing depression on stress PredictorUnstandardized Coefficient Standard error Standardized Coefficient tsig Stress

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Multiple Correlation

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How well do they predict depression together? depression loneliness R2R2

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How well do they predict depression together? depression stress R2R2

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Reasons for Multiple Regression 1) It allows you to directly compare the effect sizes for different predictor variables 2) Adding additional predictors that are related to your Y variable (we call them covariates) allows you to explain more of the residual variance. This makes MS error smaller and increases your power. 2) If you are worried that your key predictor is confounded with other variables, you can “partial them out” or “control for them” in your multiple regression by including them in the analysis. depression loneliness stress (a) (b) (c)

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Combining Types of Predictors T-tests and ANOVAs use group variables to predict continuous outcomes Correlations and simple regressions use continuous variables to predict continuous outcomes Multiple regressions allow you to use 1) information about group membership and 2) information about other continuous measurements, in the same analysis

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Combining Types of Predictors WHY would we want this? Imagine that we have a control group and a highly-provoked group, and we also measure the “TypeA-ness” of each participant. We noticed that because of streaky random sampling, we got more TypeA people in the control group than in the provoked group. Multiple regression allows us to see if there was an effect of our manipulation, controlling for individual differences in TypeA-ness. Basically, it allows us to put a situational manipulation and a personality scale measurement into the same study.

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All of the techniques we’ve covered so far can be expressed as special cases of multiple regression If you run a multiple regression with an intercept and no slope, the t-test for the intercept is the same as a single sample t-test. If you put in a dichotomous (0/1) predictor, the t-test for your slope will be the same as an independent samples t-test. If you put in dummy variables for multiple groups, your regression ANOVA will be the same as your one-way ANOVA or two-way ANOVA. If you put in one continuous predictor, your β will be the same as your r.

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General Linear Model Plus multiple regression can do so much more! Looking at several continuous predictors together in one model. Controlling for confounds. Using covariates to “soak up” residual variance. Looking at categorical and continuous predictors together in one model. Looking at interactions between categorical and continuous variables.